Download Correlation2

Digression - Hypotheses
• Many research designs involve statistical tests –
involve accepting or rejecting a hypothesis
• Null (statistical) hypotheses assume no relationship
2
between two or more variables.
• Statistics are used to test null hypotheses
1
3
– E.g. We assume that there is no relationship between
weight and fast food consumption until we find
statistical evidence that there is
Probability
4
1
• Probability is the odds that a certain event will
occur
• In research, we deal with the odds that patterns in
data have emerged by chance vs. they are
2
representative of a real relationship
• Remember – inference is the key…samples and
populations
3
• Alpha (α) is the probability level (or significance
level) set, in advance, by the researcher as the
odds that something occurs by chance
Probability
• Alpha levels (cont.)
1
– E.g. a = .05 means that there will be a 5%
chance that significant findings are due to
chance rather than a relationship in the data
2
Probability
1
• Most statistical tests produce a p-value
that is then compared to the a-level to
accept or reject the null hypothesis
• E.g. Researcher sets significance level at
.05 a priori; test results show p = .02.
• Researcher can then reject the null
hypothesis and conclude the result was not
due to chance but to there being a real
relationship in the data
• How about p = .051, when a-level = .05?
1
Error
• Significance levels (e.g. a = .05) are set
in order to avoid error
2
3
– Type I error = rejection of the null
hypothesis when it was actually true
• Conclusion = relationship; there wasn’t one
(false positive) (= a)
4
– Type II error = acceptance of the null
hypothesis when it was actually false
• Conclusion = no relationship; there was one
Error – Truth Table
1
Null True
Null False
4
2
Accept

Type II error
Type I error

3
Reject
Back to Our Example
• Conclusion: No relationship exists between
weight and fast food consumption with this
group of respondents
1
Really?
1
2
• Conclusion: We have found no
evidence that a relationship exists
between weight and fast food
consumption with this group of subjects
– Do you believe this? Can you critique it?
Construct validity? External validity?
– Thinking in this fashion will help you adopt
a critical stance when reading research
1
Another Example
• Now let’s see if a relationship exists
between weight and the number of
piercings a person has
– What’s your guess (hypothesis) about how
the results of this test will turn out?
– It’s fine to guess, but remember that our
null hypothesis is that no relationship
exists, until the data shows otherwise
Another Example (continued)
• What can we conclude from this test?
1
2
• Does this mean that  weight causes 
piercings, or vice versa, or what?
Correlations and causality
1
•
•
•
Correlations only describe the
relationship, they do not prove cause and
effect
Correlation is a necessary, but not
sufficient condition for determining
causality
There are Three Requirements to Infer a
Causal Relationship
Causality…
1
 A statistically significant relationship
between the variables
 The causal variable occurred prior to the
other variable
 There are no other factors that could
account for the cause

Correlation studies do not meet the last
requirement and may not meet the second
requirement (go back to internal validity –
497)
Correlations and causality

If there is a relationship between weight
and # piercings it could be because

2


3

weight  # piercings
1
weight  # piercings
weight  some other factor  # piercings
Which do you think is most likely here?
4
Other Types of Correlations
1
• Other measures of correlation between
two variables:
– Point-biserial correlation=use when you
have a dichotomous variable
• The formula for computing a PBC is actually
just a mathematical simplification of the formula
used to compute Pearson’s r, so to compute a
PBC in SPSS, just compute r and the result is
the same
1
Other Types of Correlations
• Other measures of
correlation between two
variables: (cont.)
– Spearman rho
correlation; use with
ordinal (rank) data
• Computed in SPSS the
same way as Pearson’s
r…simply toggle the
Spearman button on the
Bivariate Correlations
window
Coefficient of Determination
1
2
3

Correlation Coefficient Squared
 Percentage of the variability among scores on
one variable that can be attributed to
differences in the scores on the other variable
 The coefficient of determination is useful
because it gives the proportion of the
variance of one variable that is predictable
from the other variable
 Next week we will discuss regression, which
builds upon correlation and utilizes this
coefficient of determination
Correlation in excel
1
Use the function
“correl”
The “arguments”
(components) of
the function are
the two arrays
Applets (see applets page)
1
•
http://www.stat.uiuc.edu/courses/stat100/java/GCApplet/GCAppletFrame.html
• http:// www.stat.tamu.edu /~west/applets/clicktest.html
• http://www.stat.tamu.edu/~west/applets/rplot.html
2